- #1
- 19,854
- 10,839
Definition/Summary
The 1-forms (or covectors or psuedovectors) of a vector space with local basis (dx_1,dx_2,\dots,dx_n) are elements of a vector space with local basis (dx^1,dx^2,\dots,dx^n)
The 2-forms are elements of the exterior product space with local basis (dx^1\wedge dx^2,\ \dots)
In flat space with a global basis, the "d"s in the bases may be omitted.
In ordinary 3-dimensional space, a 2-form is a directed area, whose normal covector (1-form) is the dual (Hodge dual) of the 2-form.
Equations
In ordinary 3-dimensional space with basis (i,j,k):
the 2-forms have the basis:
(j\wedge k,\ k\wedge i,\ i\wedge j)
and the 3-forms are all multiples of:
i\wedge j \wedge k
and there are no higher forms.
The curl \mathbf{\nabla}\times\mathbf{a} of a vector and the cross product \mathbf{a}\times\mathbf{b} of two vectors are covectors, or 1-forms, whose duals (Hodge duals) are 2-forms which are, respectively, the exterior derivative and exterior product of their covectors:
\ast(\mathbf{\nabla}\times\mathbf{a}_i)\ =\ d \mathbf{a}^i
\ast(\mathbf{a}_i\times\mathbf{b}_i)\ =\ \mathbf{a}^i\wedge \mathbf{b}^i
Extended explanation
p-forms (differential forms):
Generally, for any number p, the p-forms are elements of the exterior product space with basis (dx^1\wedge dx^2\wedge\cdots \wedge dx^p,\cdots )
p-forms in 4-dimensional space-time:
The 2-forms in 4-dimensional space-time (Newtonian or Einsteinian) with basis (t,i,j,k) have the basis:
(t\wedge i,\ t\wedge j,\ t\wedge k,\ j\wedge k,\ k\wedge i,\ i\wedge j)
and the 3-forms have the basis:
(i\wedge j \wedge k,\ t\wedge i \wedge j,\ t\wedge j\wedge k,\ t\wedge k\wedge i,)
and the 4-forms are all multiples of:
t\wedge i\wedge j \wedge k
and there are no higher forms.
Electromagnetic 2-forms
The best-known 2-forms are the Faraday 2-form for electromagnetic field strength \mathbf{F}\,=\,\frac{1}{2} F_{ij}dx^i\wedge dx^j, with coordinates (E_x,E_y,E_z,B_x,B_y,B_z), and its dual (Hodge dual), the Maxwell 2-form \ast\mathbf{F}, with coordinates (-E_x,-E_y,-E_z,B_x,B_y,B_z)
Maxwell's equations may be written:
d \mathbf{F}\ =\ 0
d(\ast\mathbf{F})\ =\ \mathbf{J}
where \mathbf{J} is the current 3-form:
\mathbf{J}\ =\ \ast(\rho,\ J_x,\ J_y,\ J_z) = \rho i\wedge j\wedge k\ +\ J_x t\wedge j\wedge k\ +\ J_y t\wedge k\wedge i\ +\ J_z t\wedge i\wedge j
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
The 1-forms (or covectors or psuedovectors) of a vector space with local basis (dx_1,dx_2,\dots,dx_n) are elements of a vector space with local basis (dx^1,dx^2,\dots,dx^n)
The 2-forms are elements of the exterior product space with local basis (dx^1\wedge dx^2,\ \dots)
In flat space with a global basis, the "d"s in the bases may be omitted.
In ordinary 3-dimensional space, a 2-form is a directed area, whose normal covector (1-form) is the dual (Hodge dual) of the 2-form.
Equations
In ordinary 3-dimensional space with basis (i,j,k):
the 2-forms have the basis:
(j\wedge k,\ k\wedge i,\ i\wedge j)
and the 3-forms are all multiples of:
i\wedge j \wedge k
and there are no higher forms.
The curl \mathbf{\nabla}\times\mathbf{a} of a vector and the cross product \mathbf{a}\times\mathbf{b} of two vectors are covectors, or 1-forms, whose duals (Hodge duals) are 2-forms which are, respectively, the exterior derivative and exterior product of their covectors:
\ast(\mathbf{\nabla}\times\mathbf{a}_i)\ =\ d \mathbf{a}^i
\ast(\mathbf{a}_i\times\mathbf{b}_i)\ =\ \mathbf{a}^i\wedge \mathbf{b}^i
Extended explanation
p-forms (differential forms):
Generally, for any number p, the p-forms are elements of the exterior product space with basis (dx^1\wedge dx^2\wedge\cdots \wedge dx^p,\cdots )
p-forms in 4-dimensional space-time:
The 2-forms in 4-dimensional space-time (Newtonian or Einsteinian) with basis (t,i,j,k) have the basis:
(t\wedge i,\ t\wedge j,\ t\wedge k,\ j\wedge k,\ k\wedge i,\ i\wedge j)
and the 3-forms have the basis:
(i\wedge j \wedge k,\ t\wedge i \wedge j,\ t\wedge j\wedge k,\ t\wedge k\wedge i,)
and the 4-forms are all multiples of:
t\wedge i\wedge j \wedge k
and there are no higher forms.
Electromagnetic 2-forms
The best-known 2-forms are the Faraday 2-form for electromagnetic field strength \mathbf{F}\,=\,\frac{1}{2} F_{ij}dx^i\wedge dx^j, with coordinates (E_x,E_y,E_z,B_x,B_y,B_z), and its dual (Hodge dual), the Maxwell 2-form \ast\mathbf{F}, with coordinates (-E_x,-E_y,-E_z,B_x,B_y,B_z)
Maxwell's equations may be written:
d \mathbf{F}\ =\ 0
d(\ast\mathbf{F})\ =\ \mathbf{J}
where \mathbf{J} is the current 3-form:
\mathbf{J}\ =\ \ast(\rho,\ J_x,\ J_y,\ J_z) = \rho i\wedge j\wedge k\ +\ J_x t\wedge j\wedge k\ +\ J_y t\wedge k\wedge i\ +\ J_z t\wedge i\wedge j
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!